temporal weisfeiler-lehman analysis

This work investigates graphs that maximize the number of iterations required by the Colour Refinement algorithm—so-called “long-refinement graphs”—with a focus on classifying all such graphs of maximum degree at most 4 and arbitrary degree. The central problem is to characterize the existence, structural properties, and termination behavior of graphs requiring the maximal possible number of refinement steps. Using reverse engineering, long-refinement graphs are encoded as compact strings; structural analysis and closure under complementation are then leveraged to achieve the first complete classification of long-refinement graphs up to maximum degree 4: except for one sporadic case, all low-degree instances belong to known families. Moreover, it is rigorously proven that no graph exists whose vertices are distinguished only in the final iteration. These results confirm the tightness of the Kiefer–McKay upper bound and settle the classification problem, thereby revealing the intrinsic complexity boundary of Colour Refinement.

This work addresses the limitations of high-order Weisfeiler–Lehman (WL) tests, which suffer from prohibitive computational complexity (e.g., O(n⁴)) and poor scalability, while the standard 1-WL test lacks discriminative power. The authors propose DRESS, a scalable graph refinement framework grounded in continuous dynamical systems, which achieves expressiveness beyond both 1-WL and even 3-WL through parameter-free dynamic equations defined on edges. Variants such as Motif-DRESS and Delta-DRESS are introduced, establishing—for the first time—a theoretical link between continuous dynamical systems and the graph reconstruction conjecture. Notably, DRESS successfully distinguishes strongly regular graphs like the Rook and Shrikhande graphs, which are indistinguishable by 3-WL, all without resorting to costly higher-order tensor operations. The method combines generality with efficiency, significantly enhancing scalability while circumventing the computational overhead of high-order WL tests.

To address the challenge of adaptively selecting temporal resolution in time-series graph visualization, this paper proposes an automated recommendation method based on zigzag persistent homology, which identifies optimal time granularity by detecting salient topological changes in graph structure. This work introduces zigzag persistent homology to the task of temporal resolution recommendation for time-series graphs for the first time. It further designs a novel timeline encoding—“colored barcodes”—to intuitively visualize multi-scale topological evolution. Additionally, a web-based prototype system supporting interactive exploration is implemented. In a user study involving 27 participants and quantitative evaluation, the method significantly improves efficiency in anomaly detection and pattern discovery for sparse time-series graphs, demonstrating advantages in both robustness and interpretability.

This work addresses the limitation of the Weisfeiler–Leman (WL) test in distinguishing Cai–Fürer–Immerman (CFI) graph pairs by proposing Δ^ℓ-DRESS, a novel method that integrates ℓ-iteration node deletion into the DRESS continuous structure refinement framework. By applying Original-DRESS to all subgraphs obtained by removing ℓ nodes and comparing the resulting output histograms, Δ^ℓ-DRESS systematically enhances discriminative power. This approach is the first to combine node deletion with DRESS, provably achieving the expressiveness of (ℓ+2)-WL: each additional deletion layer elevates the method’s capability by one WL level. Empirically, Δ^ℓ-DRESS operates in polynomial time for fixed ℓ, with Δ⁰ through Δ³ matching 2-WL to 5-WL expressiveness, successfully distinguishing CFI(K₃) through CFI(K₆) but failing on CFI(K₇), thereby validating the theoretical boundary.

This paper establishes a tight lower bound on the minimum number of iterations required by the $k$-dimensional Weisfeiler–Leman ($k$-WL) algorithm to decide graph isomorphism. Addressing the long-standing open question—posed in 2001—of whether $k$-WL iteration complexity can exceed linear time, the authors construct a family of CFI graphs and combine WL coloring dynamics analysis with logical characterizations of graphs. They prove that, for general $n$-vertex graphs, $Omega(n^{k/2})$ iterations are necessary and sufficient for $k$-WL to achieve full coarsest equitable partition refinement. This resolves the open problem definitively. Moreover, the result yields the first quantitative connections between WL iteration depth and fundamental concepts in theoretical computer science: definability in first-order logic with counting, expressive power of graph neural networks, and computational complexity. The work thus provides a foundational theoretical basis for graph representation learning.

Traditional message-passing networks and their simplicial complex extensions struggle to distinguish mesh structures with identical connectivity but differing geometric embeddings, due to a lack of geometric awareness. This work proposes the Geometric Simplicial Weisfeiler–Lehman (GSWL) test, which for the first time incorporates vertex coordinates into the color refinement process, thereby establishing a geometric-aware simplicial message-passing framework. By integrating an Euler characteristic transform, the framework’s expressive power is fully characterized. We prove that GSWL is equivalent to geometric-aware message passing over families of finite geometric simplicial complexes, thus constructing a hierarchical theory bridging combinatorial and geometric expressivity. Experiments on both synthetic and real-world mesh data demonstrate that the proposed approach substantially enhances expressive power, clearly revealing the modeling advantages conferred by explicit geometric information.

This work investigates the homomorphism counting problem for temporal patterns with partially ordered edges in large temporal graphs, aiming to characterize the structural expressiveness of temporal graphs. By establishing an equivalence between temporal graph isomorphism and homomorphism counts of temporal patterns, the study formulates a temporal analogue of Lovász’s isomorphism theorem. It introduces a novel width parameter, termed *toadwidth*, to analyze fixed-parameter tractability. Combining techniques from parameterized complexity, extensions of clique-width, and combinatorial graph theory, the paper proves that two temporal graphs are isomorphic if and only if they admit identical homomorphism counts for all temporal patterns; that homomorphism counting is fixed-parameter tractable when parameterized by bounded toadwidth; and provides a sharp dichotomy criterion for the parameterized complexity of homomorphism counting in the case of totally ordered temporal patterns.

Traditional 1-WL stable coloring algorithms struggle to scale to web-scale graphs due to their inherently sequential nature and high memory overhead. This work addresses this limitation by leveraging the linear-algebraic formulation of 1-WL coloring to propose the first randomized refinement algorithm with rigorous probabilistic correctness guarantees. Furthermore, we introduce a provably correct graph batching strategy that preserves coloring accuracy while enabling efficient mapping of computations onto GPU primitives. By integrating randomized algorithms, graph partitioning, and CUDA-based parallelism, our approach successfully achieves stable coloring on web-scale graphs with over 30 billion edges—attaining up to two orders of magnitude speedup over CPU baselines, whereas conventional methods either time out or fail entirely at this scale.

Existing methods struggle to effectively model predictive local interaction motifs—such as reciprocation, repetition, star patterns, and triadic flows—in temporal graphs. This work proposes a compact, leakage-free 13-dimensional temporal motif feature mapping that can be linearly embedded into any static or temporal graph neural network without modifying its original architecture, thereby enhancing its sensitivity to local temporal patterns. Leveraging the observation that motif activity organizes along three stable axes across scales, we construct a universal set of candidate features and characterize their expressive power within the temporal Weisfeiler–Leman framework. Experiments demonstrate consistent performance gains across diverse tasks—including link property prediction, edge classification, and graph-level classification—when integrating our motif representation into five baseline models.

该研究提出了一种基于根树框架的方法，通过改进的LCA算法和BFS遍历，在线性时间内检测基因图中的超泡结构。